Students’ Measurement Experiences and Responses to

Length Comparison, Seriation and Proportioning Tasks

Prajakt P. Pande* & Jayashree Ramadas**

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India*prajaktp@hbcse.tifr.res.in, **jr@hbcse.tifr.res.in

Measurement is central to the practice of science and should be to science learning as

well. Seen from the perspective of science, measurement essentially involves movement

from qualitative to quantitative accounts of phenomena. We propose that qualitative

experiences of attributes of objects could develop into quantitative measures through

some intermediate steps like comparison, seriation and use of a referent. We observed 6

students’ measurement experiences and responses to a sequence of questions and tasks

on length measurement, starting from experiencing the attribute qualitatively to

quantifying it through these steps.

Introduction

Difficulties in the teaching and learning of measurement are well documented in the mathematics educationliterature (Hiebert, 1984; Bragg & Outhred, 2000; Barrett, Jones, Thornton & Dickson, 2003). This researchproblematizes measurement as an adult skill to be taught to students while taking account of their difficulties inperforming the required tasks. Historically however, measurement arose out of certain needs, admittedly of theadult world, which eventually helped shape the development of science. The underlying motivation, whetherarising from commerce, communication or science, was to move from a qualitative understanding of the world toa quantitative one. We suggest that in children too this transition should be seen from a developmentalperspective, driven by real-world experiences and actions on the world. The Piagetian concepts of classification,comparison and seriation are used in this paper to frame pre-measurement tasks for students in the primaryschool. The students' responses are interpreted in terms of epistemic actions.

Measurement in Science

Quantification and measurements are central to discoveries and inventions in science and technology.Measurements are important in testing scientific theories or proposing new ones (Kuhn, 1961). Quantitativetechniques bring uniformity and universality to the data and their interpretations, reduce distances incommunication and thus, contribute objectivity to the practice of science (Porter, 1995).

Scientific theories and explanations have not always been quantitative. Quantification of time, space and weightemerged out of practical demands that forced an attention to numerical measurements and calculations (Crombie,1961). For Aristotle, ‘qualitative’ and ‘quantitative’ were distinct categories of phenomena and his explanationswere largely based on classification. Although Aristotle predicted some quantitative relationships, the predictionsdid not aim for measurement and calculation. Grosseteste and Bacon in the 13th century began characterizing theAristotelian ‘nature’ mathematically. Medieval Platonists, unlike Aristotle, looked for explanations not inimmediate experiences but in theoretical concepts capable of quantification. In the 16th century, Galileo’scontributions to measurement through his experiments set a milestone in the history of measurement in science(Crombie, 1961). Kepler, Galileo and later Newton contributed to the dynamics that has influenced, to a greatextent, the nature of physical science.

Gerard (1961) describes how historically scientific practices in biology moved towards quantification. Senseexperiences give us direct qualitative understanding of the objects around. We tend eventually to classify thosesensed objects into categories (analogous with observation and examination of morphology of organisms inbiology followed by their taxonomic classification). Classification is followed by quantification of staticattributes of objects and then the dynamic ones, through measurement (Gerard, 1961).

Guerlac (1961) documents that early in the field of chemistry, the qualitative properties of substances were usedin order to classify them. Measurements became important relatively later in classification of substances and incharacterizing relationships, first between substances and elements and then between elements themselves.

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Guerlac sees a parallel in the qualitative to quantitative movement in science and science curricula. Students,while learning chemistry, usually learn about qualitative analysis of substances followed by a detailed and moredelicate quantitative analysis involving direct as well as indirect measurements of amounts (Guerlac, 1961).

In the history of science and in the work of scientists, these transitions happen both from qualitative toquantitative and vice versa (Gerard, 1961). Kuhn critiques the textbook view of science as a process of matchingtheoretical predictions with experimental measurements. Rather he finds a dynamic relationship between thetwo, showing how qualitative research, both empirical and theoretical, precedes fruitful quantification, which inturn helps refine a theory or propose a new one.

From ‘qualitative to Quantitative’: A Parallel tn Science Learning

In the study of the natural world, quantification is typically introduced only late in the school years. In India, theHomi Bhabha Curriculum for Primary Science (Ramadas, 1998, 2001; Vijapurkar, 2004) is exceptional inconsidering quantitative thinking as a major aim of the science curriculum and introducing measurementactivities within the context of understanding the natural world. After NCF (2005) measurement concepts weregiven an important place in the Indian school mathematics curriculum. Students at classes 2, 3 and 4 now movefrom qualitative experiences pertaining to lengths/distances to their quantification where they also encountervarious concepts underlying length measurement like visual comparison, seriation, partitioning, unit iteration,association of number with space, etc. Measurement concepts are also introduced in the science curriculum butnot yet as an essential aspect of learning and doing science (Pande, 2012, unpublished).

Children’s understanding of length measurement and the concepts underlying it has been studied from theperspective of mathematics education. The approach has been to diagnose the difficulties that students have inperforming measurement tasks and then to develop remedial instructional sequences. This research has focusedon difficulties of the transition from use of non-standard to standard units of measurement and on the gapsbetween procedural and conceptual understanding (e.g. Hiebert, 1981, 1984; Bragg & Outhred 2000, 2004;Stephan & Clements, 2003). It has not addressed the real-world motivation for measurement, i.e. to quantifyone’s observations, and to move from qualitative experiences to their quantitative understanding. Further, lengthmeasurement is rarely connected with development of the child’s understanding of space and number. It is in thisregard that a Piagetian perspective is helpful.

Developmentally children begin with qualitative notions of near and far, small and big or tall and short. Aninfant’s understanding of space develops through an interaction between visual and kinesthetic-tactileexperiences. These experiences are gained via navigation in the world and actions on it (Piaget & Inhelder, 1956;Newcombe & Learmonth, 2005). During the same period and through roughly similar kinds of interactions butsupported by schooling, children are developing the concept of number. Measurement concepts related to length,size and distance require an integration of understanding of space with the concept of number.

From a broadly Piagetian perspective, the route from qualitative comparisons to quantitative measurement hascrucial intermediate steps. The first is the progression from comparing just two quantities to comparing severalquantities, as carried out in a seriation task like placing a number of sticks in increasing or decreasing order oftheir lengths. Qualitative seriation, when imagined to extend to a very large number of quantities mayapproximate small equal increments and hence come closer to quantitative measurement. The other intermediatestep towards quantification in measurement is where one repeatedly compares the length of some standard orarbitrary unit with the length of an object. The object to be compared is estimated in terms of multiples of theunit (the referent) leading to a numerical measure, say 3 units, 4 units, etc. Seriation also, if it involves a regularincrement of such a fixed unit or referent, leads to quantitative measurement. Piaget (1952) considersclassification and seriation to be precursors to the cardinal and ordinal properties of numbers.

Piaget however does not examine closely the gestures and actions that occur in the process of acquiring theseconcepts through real world experiences. For this, the perspective of epistemic actions is helpful. Kirsh (1995)gives a fine grained description of how we think, plan and execute everyday spatial tasks by identifying spatialarrangements that physically constrain actions. The Piagetian and epistemic action perspectives together help uscharacterize the progression, via the child's actions, from qualitative experiences of the natural world to thequantification of specific attributes of this world, which is precisely the aim of measurement in science.

Developmental stages proposed by Battista (2007) and learning progressions and learning trajectories describedby Clements and Sarama (2009) are based on movement from qualitative to quantitative, although they do notuse the seriation concept, nor do they closely look at the child's actions. Battista (2007) proposed two stages ofreasoning. Non-measurement reasoning which involves visual-spatial inferences like direct or indirectcomparisons, imagined transformations or geometric properties (e.g. identifying 'shorter' and 'taller', smaller &bigger etc.) and measurement reasoning involving iterations and understanding proportions.

Students’ Measurement Experiences and Responses to Length Comparison, Seriation and Proportioning Tasks 147

Aims Of The Study

We propose that some intermediate steps are needed in moving from qualitative experience of an attribute of anobject to quantifying that attribute, which are comparison, seriation and use of a referent. Further we have alsoconsidered partitioning of continuous object, iterating unit/referent, proportionality and association of numberwith space (Piaget et al., 1981; Stephan & Clements, 2003) in framing the tasks. Following are the aims of thisstudy.

To explore:

• Students’ familiarity with measurement and contexts of measurement

• Students’ strategies in length comparisons: do they spontaneously use any referents?

• Strategy for retrieving longer and shorter things: is seriation used spontaneously?

• Students’ performance on seriation tasks, specifically their use of epistemic actions

• Students’ estimation, partitioning and proportioning abilities

Sample

Six students were selected of whom 2 girls (G1 & G2) and 2 boys (B1 & B3) had passed class 3, one boy (B2)had passed class 4 and one girl (G3) had passed class 5. The participant students lived in Dharavi recognized asone of the largest slums in India. They attended the Dharavi Transit Camp School which runs five differentmedia of instruction in one building. Students G1, G3, B1, B2, B3 studied in the Urdu medium and student B2English medium. The interviews were conducted during the summer vacation, in a room outside of the schoolpremises.

Data Collection

The students were interviewed individually in two sessions of 10-15 minutes each. Student B2 could notparticipate in the second interview session due to some logistical reasons.

In the first session students were asked to recite numbers and asked them where they used counting. Studentswere then asked about their ideas of measurement, the everyday contexts in which they used measurement andwhether they thought measurement was important. The second session focused on length measurement,proceeding from length comparisons and seriation with everyday objects (9 questions) to seriating a set of 10cardboard strips of increasing length from 2cm to 20cm in steps of 2 cm. The seriation tasks using strips (6questions) progressively increased the number of strips from 2 to 10, adding 1 or 2 strips at a time. Finally,students were given two strips, such that the length of the longer strip was in multiples of the shorter strip andwere asked, 'यह पट्टी (e.g. 4cm) इस पट्टी (e.g. 2cm) से कितनी लम्बी है?' ('How much longer is this strip than theother strip?' (7 questions, varying ratios of lengths of strips).

The entire conversation happened in Hindi. Interviews were audio recorded and transcribed. They weresupported by written notes.

Findings

Reciting of numbers and contexts of counting

Only student G3, who had passed class 5, could count till 100; even beyond 1000 according to her. Student B2who had passed class 4 could count up to 50. The other four students could count till about 15 or 20 only. All thestudents were familiar with counting money, rupee notes and coins. Except B1, all mentioned use of countingruns and players in cricket. B1 and B3 believed that counting is related to tables in mathematics. G2 believedthat counting was essential so that people do not cheat us while transacting.

Daily contexts of measurement

Students were familiar with the word ‘नापना’ (‘measurement’). All the six students said that they knew aboutlength, height and weight measurement. However, three of them B1, G1 and B3 confused the measure of heightto be their age, reporting their past or present height in terms of numbers like 8, 11, etc. The other three studentsdid not respond to the question. Three of the students said that ‘Bournvita’ or ‘Complan’ (nutrition and energy

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drinks) were required to grow in height, while one of them thought that with increasing age these drinks neededto be consumed in larger quantities.

Responding to the question, ‘Have you observed anybody doing measurements?’ (क्या आपने कभी किसी को कुछनापते हुए देखा है या खुद कुछ नापा है?), student G3 said that her father was a tailor and she often visited his shop andcarried out measurements of costumers’ body parts and clothing materials using an inch tape. Other students toohad observed length measurement carried out by a tailor. One student mentioned distance between two suburbantrain stations but could not say how it was measured. All the students knew of measurement of weight andamount (meat, vegetables and grains). Student B2 demonstrated with gestures, the use of different ‘बाट’ (standardweights used by vendors) to weigh smaller amounts (20g, 50g) and larger amounts (1kg, 2kg). G1 mentionedestimation of amount of iron and other raw material required while building a house in relation to the size of thehouse.

Students believed that people need to measure things in order to sell or buy them. Some of the students believedthat counting and measurements were necessary so that people do not cheat while they communicate.

Performance in Comparison, Seriation, Partitioning and Proportion Tasks

Comparison of lengths

When asked to name ‘long’ objects, students gave fairly non-specific responses like ‘animal’, ‘building’,‘bamboo’, ‘sky’ and elephant. Examples of ‘short’ objects were somewhat more specific like ‘younger sister’,‘stool’, ‘nail’, ‘lice’, ‘fan’, ‘ant’ and ‘crushed stone’. When asked why they had categorized these objects as‘long’ or ‘short’, most gave no responses or said it was ‘just so’. Only G2 responded that the bamboo was longerand the fan was shorter than her. Interestingly, all the four ‘long’ objects listed by all of the five students werelonger than themselves, while the five ‘short’ objects were shorter than themselves. Thus students appeared to beimplicitly/unknowingly using their body dimensions as referent for determining ‘long’ and ‘short’. Althoughgesture data was not recorded in this study, we note that our arm span, which is used to gesture size, isapproximately equal to our height.

When explicitly asked to name things longer or shorter than themselves, all except B3 predominantly mentionedpeople, family members and friends. Remarkably, when asked to repeat names of the objects that they hadmentioned, all of the students retrieved the objects in either increasing or decreasing order of length.

Seriation strategies and actions

All the students except G2 assumed an imaginary baseline while placing and arranging the strips. All seriated atleast five strips at a time but as the number of strips gradually increased, the students B1, G1, G2 and G3 foundit difficult to compare the strips and seriate them, and confusions started arising. Their new strategy involvedpicking up a few strips of comparable length, identifying either the longest or the shortest strip from that group,and identifying another strip from the same group to place next to it, thus comparing two strips at a time. Thisstrategy is similar to clustering, an epistemic action described by Kirsh (1995). Student G3 also exhibited anadditional step of physically categorizing the strips prior to picking and comparing. B3 successfully seriated allthe 10 strips by visual inspection without the intermediate action steps.

Partitioning and proportion strategies and actions

Students were given two strips (length of the longer strip in multiples of the shorter (twice, thrice, 4 times and 10times) and were asked how much longer the longer strip was than the shorter one. Students B3 and G3approached this task most systematically, aligning the two strips and iterating the shorter strip along the longerone, thus arriving at the correct number (although student G3 in each task attributed the unit ‘inch’ to the shorterstrip and responded with the respective number followed by “inch”. B3 responded with almost the exactnumbers followed by the word ‘गुना’ (‘times’). Figure 1 depicts B3’s strategy in this task.

Students’ Measurement Experiences and Responses to Length Comparison, Seriation and Proportioning Tasks 149

Figure 1. Student B3 iterating the shorter strip along the longer strip

Students B1, G1 and G2 failed to iterate the referent (here the shorter strip) along the longer one. They initiallyused some repeated gesture along each of the two strips, appearing to use an imaginary arbitrary unit, andcounted the number of segments made by these imaginary marks. Figure 2 depicts measurement strategy used bystudent B1.

Figure 2. Partition and counting strategy of student B1

Figure 3. Student G2 indicating ‘this much’ longer using gesture

A little later during the same task both B1 and G2 correctly aligned the two strips and indicated the space on thelonger strip that was not covered by the shorter one saying that the longer strip was ‘ this much’ longer than theshorter strip. G2’s strategy in this task can be seen in Figure 3 above.

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Figure 4. Student G1 partitioning the longer strip with shorter strip aligned horizontally

Student G1 mentioned that she was imagining the unit marks on a ruler and trying to make similar ones on thestrips in order to measure it. Figure 4 shows G1’s strategy of measurement. Here, student G1 perhaps exhibitedmeasurement influenced by what Stephen and Clements (2003) call the “mental representation of a ruler”.

Summary

Overall performance on the tasks was below what could be expected from students of age 8-12 years. Thecounting capability of the four class 3 students was only up to 15-20. The daily measurement contexts werelimited to tailoring, although one girl had hands-on experience of this task. Spontaneous use of self reference(comparing objects with own body dimensions) was a common strategy in length comparison tasks. Familymembers and friends were the preferred choices for comparison of heights with self. Length seriation was animplicit strategy used for retrieval and perhaps for remembering objects too. In the seriation tasks as the numberof strips to be seriated at a time increased (which in turn increased the cognitive load), a suitable strategy tocounter this load was devised by these students: i.e. to break up the task into steps, thus restructuring the taskenvironment. Restructuring often reduces the cost of search to make objects easy to notice, identify, select andplace (Kirsh, 1995). Partitioning and counting strategies were spontaneously used by the three younger studentswhile one younger and one older student used iteration to arrive at a proportionality statement.

Concluding Remarks

Mathematics education research has often focused on the transition from using non-standard to standard units.Yet it has been found advantageous to introduce the ruler early and to use both the standard and non-standardunits of measurement simultaneously (Nunes, Light & Mason, 1995; Clements, 1999; Boulton-Lewis, Wilss &Mutch, 1996). This paradox is resolved if we look at measurement not as a mathematics education problem but ascience education one, i.e. one of developing a progressively more refined understanding of the real world.Consequently, we suggest that the issue to be focused on is not simply one of developing a skill or a competencybut rather of enabling a transition from qualitative experiences to their quantitative understanding. In thistransition the crucial steps of comparison and seriation are of prime importance to help in teaching and learningof measurement. Action strategies in this regard should be helpful in studying the cognitive processes related tomeasurement.

References

Barrett, J., Jones, G., Thornton, C. & Dickson, S. (2003). Understanding Children’s Developing Strategies andConcepts for Length. In D. Clements and G. Bright (Eds.), Learning and Teaching Measurement: 2003 Year-

book, 17-30, Reston VA: NCTM.

Battista M. T. (2007). The Development of Geometric and Spatial Thinking. In Frank K. Lester, Jr. (Ed.). Second

Handbook of Research on Mathematics Teaching and Learning, 843-908.

Boulton-Lewis, G, Wilss, L. and Mutch, S. (1996). An analysis of young children's strategies and use of devisesfor length measurement, Journal of Mathematical Behavior, 15, 329-347.